ADC Resolution and Sampling Formulas

ADC formulas: the size of one step from the bits and the reference, the count for a voltage, the ideal signal to noise ratio, ENOB and the Nyquist limit.

The formulas

N is the number of bits, Vref the reference voltage and fmax the highest frequency in the signal. The step formula is for an ideal converter with a unipolar range of 0 to Vref. A real converter has offset, gain and noise errors that are on its data sheet, and the SINAD (the signal to noise and distortion ratio in dB) gives the effective number of bits. Frequencies above half the sampling rate fold back into the band, so a low-pass filter before the converter is needed, and the sampling rate is usually set well above the Nyquist limit.

Worked example: 12 bits and a 3.3 V reference

1. LSB = 3.3 / 4096 = 0.8057 mV
2. A reading of 1.65 V is 1.65 / 0.8057 mV = 2,048 counts, which is half scale
3. SNR = 6.02 × 12 + 1.76 = 74 dB

Worked example: a 4 to 20 mA loop on 250 Ω with a 5 V reference

The loop gives 1 V at 4 mA and 5 V at 20 mA.

1. LSB = 5 / 4096 = 1.221 mV
2. 4 mA (1 V) is 819.2 counts and 20 mA is 4096, so the span uses 3,277 counts
3. One count is 16 mA / 3,277 = 4.883 µA

Worked example: ENOB and the sampling rate

1. A SINAD of 68 dB gives ENOB = (68 − 1.76) / 6.02 = 11 bits
2. A signal with components up to 1 kHz needs fs above 2 kHz

Converters with a 3.3 V reference

Frequently asked questions

Does more bits always mean better accuracy?

No. The noise, offset and gain error of the converter and its reference limit the accuracy, so the number of bits only gives the resolution. The ENOB shows what the converter can really do.

What is the Nyquist limit?

A signal must be sampled at more than twice its highest frequency to be rebuilt. Anything above half the sampling rate shows up as a false lower frequency.

Why use a low-pass filter before the ADC?

It removes the frequencies above half the sampling rate that would otherwise fold into the measurement as aliasing.

Can I measure a sensor that uses only part of the range?

Yes, but it uses fewer counts, so the resolution in the sensor units is worse, as in the 4 to 20 mA example. A gain stage that fills the range gives more counts.

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